Volume of a Prism Textbook Exercise - Corbettmaths - Free Printable

Let's solve Question 1: Calculate the volume of each prism below.

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🔷 Formula for Volume of a Prism:

$$
\text{Volume} = \text{Area of Cross-Section} \times \text{Length (or Height)}
$$

The cross-section is the 2D shape at the end of the prism (e.g., rectangle, triangle, trapezium), and we multiply its area by the length of the prism (the distance it extends).

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We’ll go through each part one by one.

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(a) Rectangular Prism

Dimensions: 3 cm × 4 cm × 6 cm
This is a rectangular prism (cuboid). The cross-section is a rectangle.

$$
\text{Area of base} = 3 \times 4 = 12\,\text{cm}^2 \\
\text{Length} = 6\,\text{cm} \\
\text{Volume} = 12 \times 6 = \boxed{72}\,\text{cm}^3
$$

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(b) Triangular Prism

Right-angled triangle base: 5 cm and 6 cm, length = 20 cm

Area of triangular cross-section:
$$
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 6 = 15\,\text{cm}^2 \\
\text{Length} = 20\,\text{cm} \\
\text{Volume} = 15 \times 20 = \boxed{300}\,\text{cm}^3
$$

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(c) Triangular Prism

Triangle base: base = 7 m, height = 5 m, length = 4 m

$$
\text{Area} = \frac{1}{2} \times 7 \times 5 = 17.5\,\text{m}^2 \\
\text{Volume} = 17.5 \times 4 = \boxed{70}\,\text{m}^3
$$

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(d) Cuboid (Rectangular Prism)

Dimensions: 1.1 m × 40 cm × 2 m
⚠️ Units must be consistent! Convert 40 cm to meters:

$$
40\,\text{cm} = 0.4\,\text{m} \\
\text{Area of base} = 1.1 \times 0.4 = 0.44\,\text{m}^2 \\
\text{Length} = 2\,\text{m} \\
\text{Volume} = 0.44 \times 2 = \boxed{0.88}\,\text{m}^3
$$

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(e) Triangular Prism

Triangle: base = 7 cm, height = 8 cm, length = 5.5 cm

$$
\text{Area} = \frac{1}{2} \times 7 \times 8 = 28\,\text{cm}^2 \\
\text{Volume} = 28 \times 5.5 = \boxed{154}\,\text{cm}^3
$$

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(f) Trapezoidal Prism

Cross-section is a trapezium with parallel sides 6 cm and 9 cm, height = 12 cm? Wait — look carefully.

Wait: the diagram shows a trapezium with height = 6 cm, and two parallel sides: 9 cm and 12 cm, and the length of the prism is 12 cm?

Actually, looking closely:

- The trapezium has parallel sides of 9 cm and 12 cm, and the height between them is 6 cm.
- The length of the prism is 12 cm (this is the direction extending out).

So:

$$
\text{Area of trapezium} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height} \\
= \frac{1}{2} \times (9 + 12) \times 6 = \frac{1}{2} \times 21 \times 6 = 63\,\text{cm}^2 \\
\text{Volume} = 63 \times 12 = \boxed{756}\,\text{cm}^3
$$

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(g) Triangular Prism

Right-angled triangle: legs = 12 cm and 13 cm? Wait — check the diagram.

It shows a right triangle with one leg = 12 cm, hypotenuse = 13 cm, so we can find the other leg.

Use Pythagoras:

$$
a^2 + b^2 = c^2 \\
12^2 + b^2 = 13^2 \\
144 + b^2 = 169 \\
b^2 = 25 \Rightarrow b = 5\,\text{cm}
$$

So the two perpendicular sides are 12 cm and 5 cm.

Area of triangle:
$$
\frac{1}{2} \times 12 \times 5 = 30\,\text{cm}^2 \\
\text{Length of prism} = 22\,\text{cm} \\
\text{Volume} = 30 \times 22 = \boxed{660}\,\text{cm}^3
$$

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(h) Trapezoidal Prism

Trapezium cross-section: parallel sides = 8 cm and 10 cm, height = 40 cm? No — wait.

Looking at the diagram:

- The cross-section is a trapezium with height = 8 cm
- Parallel sides: 40 cm and ? Wait — no.

Wait: the length of the prism is 40 cm, and the cross-section is a trapezium with:

- Two parallel sides: 8 cm and 10 cm
- Height of trapezium = 8 cm? But labeled as "8 cm" vertical arrow — yes.

But actually, the height of the trapezium is 8 cm, and the length of the prism is 40 cm.

Wait — let’s clarify:

- The trapezium has:
- Parallel sides: 8 cm and 10 cm
- Height between them: 8 cm → but that seems odd — maybe not.

Wait — look again: the figure shows arrows indicating the height of the trapezium is 8 cm, and the prism extends 40 cm.

So:

$$
\text{Area of trapezium} = \frac{1}{2} \times (8 + 10) \times 8 = \frac{1}{2} \times 18 \times 8 = 72\,\text{cm}^2 \\
\text{Length} = 40\,\text{cm} \\
\text{Volume} = 72 \times 40 = \boxed{2880}\,\text{cm}^3
$$

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(i) Triangular Prism

Triangle: base = 11 cm, height = 9 cm, length = 15 cm

$$
\text{Area} = \frac{1}{2} \times 11 \times 9 = 49.5\,\text{cm}^2 \\
\text{Volume} = 49.5 \times 15 = \boxed{742.5}\,\text{cm}^3
$$

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✔ Final Answers:

| Part | Volume |
|------|--------|
| (a) | $72\,\text{cm}^3$ |
| (b) | $300\,\text{cm}^3$ |
| (c) | $70\,\text{m}^3$ |
| (d) | $0.88\,\text{m}^3$ |
| (e) | $154\,\text{cm}^3$ |
| (f) | $756\,\text{cm}^3$ |
| (g) | $660\,\text{cm}^3$ |
| (h) | $2880\,\text{cm}^3$ |
| (i) | $742.5\,\text{cm}^3$ |

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🔍 Summary of Key Steps:

1. Identify the cross-sectional shape (triangle, rectangle, trapezium).
2. Find the area of that shape.
3. Multiply by the length of the prism (the dimension going out from the cross-section).
4. Watch units — convert if needed.

Let me know if you'd like this in a printable format or with diagrams!